Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions

: In this article, we study the following fractional Schrödinger-Poisson system:

) satisfies some lower order pertur- bations, we show that there exists a constant > ε 0 0 such that for all ∈ ε ε 0, 0 ( ], the above system has a semiclassical Nehari-Pohozaev-type ground state solution v ˆε.Moreover, the decay estimate and asymptotic behavior of v ˆε { } are also investigated as → ε 0. Our results generalize and improve the ones in Liu and Zhang and Ambrosio, and some other relevant literatures.

Introduction and main results
In this article, we investigate the existence and concentration of solutions to the following fractional Schrödinger-Poisson system: where > ε 0 is a small parameter, < < t s 0 , 1, + > = − t s 2 2 3,2* s s 6 3 2 is fractional critical exponent in 3 , and −Δ α ( ) , with = α s is the fractional Laplacian operator, which can be defined as, for any → u : 3  belonging to the Schwartz class, where P.V. represents the Cauchy principal value and C s 3, ( ) stands for a normalizing constant; see [16].When = = s t 1, system (1.1) reduces to the following classical Schrödinger-Poisson system: It characterizes systems of identical charged particle systems interacting with one another in the event that magnetic field effects are negligible.In particular, a standing wave is the solution for such a system.We cite [9] for a more comprehensive explanation of this system.For Schrödinger-Poisson systems, concerning existence, nonexistence, and multiplicity for both bound states and ground states, we cite [7,19,35,45,48].For equation (1.2), with ) and some intervals contain μ, Ruiz [35] demonstrated that equation (1.2) admits a positive radial solution by building a constrained minimization on a new manifold based on the Nehari manifold and the Pohozaev identity.In circumstances where − + u f u ( ) satisfies Berestycki-Lions condition, Azzollini et al. [7] used the penalization method and the Ljusternik-Schnirelmann theory to show that equation (1.2) admits nontrivial solutions.Lately, there are some studies on the semiclassical state of the system (1.2) under various potential V and nonlinearity f conditions.As an illustration, He [20] investigated the multiplicity and concentration of positive solutions and showed that positive solutions concentrate around the global minimum of the potential V in the semiclassical limit, but the nonlinearity term satisfies subcritical growth.For the critical case, Chen et al. [12] employed a constrained minimization on a Nehari-Pohozaev manifold to show that system (1.2) admits a semiclassical ground state solution and study properties of these ground state solution, such as convergence and decay estimate.He and Zou [21] proved that system (1.2) admits a positive ground state solution concentrating around the global minimum of the potential V , and they also investigated the exponential decay of ground state solutions.In [8,23,25,26,33,34,36,39,49] and the references therein, additional results regarding semiclassical states and variational method on elliptic equations are provided.
If = ϕ x 0, ( ) system (1.1) becomes the fractional Schrödinger equation like The fractional Schrödinger equation has standing wave solutions in the form of equation (1.3): i.e., solutions of the form = − ∕ ψ x t e u x , iEt ε ( ) ( ), where E is a constant and u x ( ) is a solution of equation (1.3).A key equation in fractional quantum mechanics is the fractional Schrödinger equation.In the last two decades, many authors widely investigated problem (1.3), we quote [4,13,17,37,38,40] and its references for the readers' information.In the case where = V 1 and f satisfies subcritical growth and Ambrosetti-Rabinowitz condition, Felmer et al. [17] investigated the existence, regularity, and symmetry of positive solutions to equation (1.3).Secchi [40] investigated equation (1.3) under reasonable assumptions on the behavior of the potential V at infinity and the nonlinearity f is superlinear with subcritical growth, but it does not satisfies the Ambrosetti- Rabinowitz condition.The existence and multiplicity results of equation (1.3) were shown by Shang et al. [37] with critical growth and needing the following global assumption on the potential V introduced by Rabinowitz [30]: By using the penalization technique and the extension method [11], Alves and Miyagaki [1] proved equation (1.3) admits positive solutions, and studied their concentration behavior when V verifies V 1 ( ) and V 2 ( ) and f satisfies subcritical growth.Later, Ambrosio [5] used penalization method and the Ljusternik-Schnirelmann theory to study the multiplicity of positive solutions of equation (1.3) with critical nonlinearities.See also [10,41] for critical problems in bounded domains and [2,3] for critical fractional periodic problems.Now, to show our results, we make the following hypotheses on V : ] is increasing on +∞ 0, ( ), and for some > ρ 0 0 , such that a .e . .For the nonlinearity f , we assume the following conditions: ), where , 0 ( ) for all ≥ τ 0.
Note that Gao et al. [18] first introduced V 2 ( ) and F 2 ( ) to study the fractional Schrödinger-Poisson system.However, in [18], the authors only showed the existence of ground state solutions for equation (1.1) with = ε 1 and subcritical growth under , and there exists ∈ q 2, 2* s ( ) and a constant > C 0 0 such that in F 3 ( ), namely, ≤ s t.As far as we know, less research has been done on systems like (1.1) except for works [6,28,29,44,47].Yu et al. in [47] considered with the fractional nonlinear Schrödinger-Poisson system: ( ) ( ) ( ) has a global maximum and is positive, and they showed the existence of a positive ground state solution and studied the concentration position of these ground state solutions as → ε 0. When the f is critical nonlinearity, Liu and Zhang [28] and Ambrosio [6]  considered the critical problem: ( ).In [28], the nonlinearity → f : is of C 1 class and satisfies the following conditions: for all > τ 0 with some > ρ 0 and ∈ − σ q 3, 1 ( ) .
Critical fractional Schrödinger-Poisson systems with lower perturbations  3 Obviously, it follows from By using minimax theorems and Ljusternik-Schnirelmann theory, they shown the multiplicity and concentration of solutions for system (1.5).Later, Ambrosio [6] improved the conditions of nonlinearity f , f is no longer C 1 -class, and satisfies the following conditions: ( They investigated the relationship between the number of positive solutions and the topology of the set where the potential reaches its minimum value utilizing penalization techniques and Ljusternik-Schnirelmann theory. Based on the aforementioned facts, we prove in this study that semiclassical ground state solutions exist for equation (1.1) with critical growth and more general subcritical perturbation.In contrast to [28], we concentrate on the analysis of equation (1.1) using a more general subcritical perturbation f with F 1 ( )-F 3 ( ).First, f is a continuous function rather than of C 1 -class, which leads to Nehari-Pohozaev manifold not being a C 1 -manifold.Second, f no longer satisfies monotonicity condition f 3 ( ), which plays a crucial role in using the Nehari manifold method.Finally, we use condition F 1 ( ), which is weaker than f 1 ( ) and f 2 ( ).Compared with [6], f no longer satisfies Ambrosetti-Rabinowitz condition ′ f 3 ( ) and monotonicity condition ′ f 4 ( ), so we have great difficulty in using Nehari manifold method; moreover, the condition F 1 ( ) is more general than ′ f 1 ( ) and ′ f 2 ( ).In a manner, our results generalize and improve the works of [6,18,28].
When we studying system (1.1) under more general subcritical perturbation f with F 1 ( )-F 3 ( ), there are three main difficulties.First, system (1.1) has two nonlocal terms makes our analysis more complicated and intriguing.Second, the nonlinearity f is not of C 1 -class, and this leads to Nehari-Pohozaev manifold not being a C 1 -manifold; we draw attention to the proofs in [35,44], which are on the basis of minimizing the associated functional restricted to a fitting manifold which is C 1 .As a result, the arguments provided by [35,44]

( ) ( ) (
] due to the domain 3 and the critical Sobolev exponent.We use the Concentration-Compactness Lemma for the fractional Laplacian, see [31].Consequently, a more thorough analysis is required, which originates from [12]. Here, we will list our main results.( ), where v ˆε satisfies the following statements: ], the function v ˆε | | achieves its maximum at a point x ε , which satisfies x x for all x and ε ε ˆ0, .) converges in H s 3 ( ) to a ground state solution u of the following autonomous equation: In order to overcome the lack of compactness of the embedding we need show the existence of positive ground state solutions to the limit equation associated with equation (1.1) where a is a positive constant with Theorem The structure of this article is as follows: In Section 2, we provide some preliminary lemmas which will be used later; in Section 3, we prove the autonomous equation (1.7) and conclude the proof of Theorem 1.3; in Section 4, we show the existence of semiclassical ground state solutions to equation (1.1) for all ∈ ε ε 0, 0 ( ]and give the proof of Theorem 1.1; in Sections 5 and 6, we study the concentration phenomenon and convergence of ground state solutions.In particular, we obtain the decay estimate of solution, which complete the proof of Theorem 1.2.

Preliminaries
Throughout this article, we denote ⋅ p ‖ ‖ the usual norm of the space ) denotes some positive constants may change from line to line.First, we introduce the space H s 3 ( ), which is defined as follows: The inner product and the norm are defined, respectively, as follows: For convenience, we set where D α, 2 3  ( ) is defined by According to the Lax-Milgram Theorem, for ∈ u H s , there is a unique ( ) that satisfies: ( ) Observe that for any > ε 0, Making the scaling = u x v εx ( ) ( ), we can rewrite the system (1.1) as the following equivalent system: , we define the functional of equation (1.7) in H s 3 ( ) as follows: We also study the counterpart results of the concentration phenomenon as → ε 0 for ground state solutions to equation (1.1).For this reason, we define the Pohozaev-type functional According to [15], any solution u of equation (1.6) satisfies = u 0 ε ( ) .Inspired by this fact and by the work of Ruiz [35], we introduce the following functional on H : and the constant potential case by We define the Nehari-Pohozaev manifold of ε by and the constant potential case For any ∈ u ε , we say that Then, every nontrivial solution of equation (2.2) is contained in ε .In particular, we call a nontrivial solution u ˆof equation (2.2) to be a ground state solution of Nehari-Pohozaev-type if Similarly, every nontrivial solution of equation (1.7) is contained in a .In particular, we call a nontrivial solution u ˜of equation (1.7) to be a ground state solution of Nehari-Pohozaev-type if By simple calculation, we obtain the following lemmas.
Proof.We know that Critical fractional Schrödinger-Poisson systems with lower perturbations  7 Lemma 2.4.Assume F 2 ( ) holds, then Proof.Without loss of generality, we can assume that ≠ τ 0 and set s t s t s t By a direct computation, we have 3 Constant potential case Lemma 3.1.Assume that F 1 ( ) and F 2 ( ) hold and Proof.By Lemma 2.4, it is simple to verify that where ).Thus, we have Clearly, as a result of F 1 ( ) and Moreover, we assert that the critical point of τ 0 is unique.In fact, we just assume that there are two points Next, we will prove > m 0 a .Indeed, it follows from F 1 ( ) that there exists a constant C such that thus for any ∈ u a , by equation (3.5), Lemma 3.2, and Sobolev inequality, we obtain Therefore, we complete the proof.□ Lemma 3.5.Assume that F 1 ( ) holds and and It follows from F 1 ( ) and the Brézis-Lieb lemma that Thus, one has Combing equations (3.9), (3.11), and (3.13), we know equation (3.7) holds, Finally, we note that Combining equations (3.7), (3.10), and (3.13), we can obtain equation (3.8).
, and , and , and ∈ x 0 3 .Based on [42], we know that this is true From Lemmas 3.3 and 3.4, there exists a > τ 0 Next, we will prove that there exist two constants . First, we claim that τ ε is bounded from below by a positive constant.Otherwise, we could find a sequence → ε 0 this is clearly a contradiction.On the other hand, by F 3 ( ), we obtain that which implies that there exists > τ* 0 such that ≤ τ τ* ε .Therefore, the claim is proved.Thus, we conclude that Next, we separate three cases: By computations, we obtain that . , for any > μ 0, we have that .

| | ( )
and for any > μ 0, we have be a minimizing sequence for m a such that a n a

( )
We divide the proof into three steps as follows: Step Step 2: There exist a sequence Suppose, by contradiction, that for all > R 0, we can conclude that Step 3: m a is achieved.Letting a , and u ˜n { } is still a bounded minimizing sequence for m a .Up to a subsequence, we can assume that there is a ∈ u H ˜s 3 ( ) such that and So we have We first claim that By contradiction, suppose that there exist > ε 0 0 and a sequence τ n { } such that Then, there exist two functions U 1 and ∈ U C which contradicts equation (3.27).According to equation (3.26), there exists > δ 0 1 such that According to Lemma 2.3 in [46], there exists a deformation ∈ × η C H H 0, 1 , By Lemma 3.
2 , which contradicts equation (3.32).In fact, we define By Lemma 3.3 and the Brouwer degree, we obtain . From Lemmas 3.7 and 3.8, we can easily obtain that a has a critical point ∈ u ˜a such that In summary, we complete the proof of Theorem 1.3.

Nonconstant potential case
In this section, we used method due to Jeanjean and Toland [24] to prove the existence of ground state solutions for equation (2.2).For this purpose, for ∈ λ , 1 [ ], we introduce the following two families of func- tionals on H s 3 ( ) defined by and Then, the following Pohozaev-type identity holds: Similarly to equation (4.3), for all ∈ λ , 1 [ ] and ∈ u H s 3 ( ), we set , and [ ], we define the following functional in H s 3 ( ): We can easily check that there exists > T 1 such that ), the following statement holds: By Lemma 2.4 and equation (2.9), we have Proof.By V 2 ( ), we obtain where C 1 is a positive constant.This implies . if we take Therefore, the proof is completed.

Letting
Critical fractional Schrödinger-Poisson systems with lower perturbations  21 We know that Proof.According to Lemma 4.2 and Theorem in [27], for almost every ∈ λ , 1 [ ] and every > ε 0, there is a bounded sequence { ( )} ( ) that as a matter of convenience, still denote it by u n { } such that as → ∞ n There exists ( ) { }; for convenience, we denote it by u such that, up to a subsequence, ( ) for all ≤ < r 2 2 * s , and → u u n a.e. in 3 .We have . By equation (4.16), we obtain and Combining with equations (4.17) and (4.18), we have , and so . By Lemma 4.5 and Fatou's lemma, it is easy for us to obtain Hence, we prove Proof.In view of equations (5.2) and (5.3), we deduce that where δ 0 is given in Lemma 5.1.
Proof.By V 2 ( ), there exist constants > ρ 0 0 such that So we obtain Proof.

∫ ∫
, then it follows from F 3 and equations (5.8) and (5.9) that as → ∞ n , , and we have where δ is given by Lemma 4.5.This is clearly a contradiction.So equation (5.13) holds.According to (5.12), (5.13), as well as the boundedness of u n { } in H s 3 ( ), we have that Proof.For ≥ r 2, it follows from Lemma 6.1 and the standard bootstrap argument (see [32]) that there exists

(
Choosing μ large enough such that the above three limits equal to +∞, e.g.,
1: We prove u n { } is bounded in H s 3 ( ).It follows that by equations (2.11) and (3.15), we have that Critical fractional Schrödinger-Poisson systems with lower perturbations  25 Critical fractional Schrödinger-Poisson systems with lower perturbations  27 Critical fractional Schrödinger-Poisson systems with lower perturbations  29 By using the Sobolev embedding theorem, we deduce that there exists >As in the proof of equation (6.16), we deduce that as → ∞ n , ˜is a ground state solution of equation (1.6), as asserted.□Proof of Theorem 1.2.In view of Theorem 1.1, the function = − , then (i) follows from equation (6.29).Moreover, 6.7 imply the validity of (iii).